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Suppose I use the following definition of a Cartier divisor: a global section of the quotient sheaf $\mathcal{K}^*/\mathcal{O}^*$.Because of the property of quotient sheaves a Cartier divisor on $X$ can be described by giving an open cover $\{U_i\}$ of $X$ and $f_i \in \Gamma(U_i, \mathcal{K}^*)$ such that $f_i / f_j \in \Gamma(U_i \cap U_j, \mathcal{O}^*)$

I thought it would be interesting to use this definition to figure out the divisors on $\mathbb{P}^1$. In pariticular, I want to show that every divisor is a multiple of the hypersurface $x_0=0$. This would reprove that $D \sim dH$.

I am getting stuck trying to figure out what the global sections of this sheaf are. However, even if I were to find the global sections I am not exactly sure how I would relate them to a hypersurface.

Would I just compute the vanishing set of the global section I find?

Its still a bit confusing to me that Cartier divisors are defined as global sections of a quotient sheaf and Weil divisors are closed subschemes of codimension $1$ of our scheme.

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Let's look at the global section of this sheaf. Element of $\mathscr K^*$ are exactly non-identically zero rational functions $\frac{f}{g}$, and $\mathcal O^*$ is simply the global section of non-identically zero holomorphic functions, i.e $\mathcal O^*(\Bbb P^1) =k^*$.

Assuming $k$ is algebraically closed, any such $f/h$ will be product of some linear factors quotiented by product of some other linear factors, i.e $f/h$ is equivalent to the data of the zeroes and the poles with multiplicities, which is exactly the data of a divisor $D = \sum_{p \in \Bbb P^1} D(p) p$. For example, the Cartier divisor $$[\frac{(x_0 - x_1)(x_0 + x_1)}{x_1^2}]$$ will correspond to the Weil divisor $D = [1:1] + [1:-1] - 2 \cdot [1:0]$.

Now, it's easy to check that if $D,D'$ have the same degree then there is a rational function $f$ with $\text{div}(f) = D - D'$. Taking $D' = \deg(D) H$ shows that any divisor is equivalent to some multiple of $H$.

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